I've seen broody hens get newly hatched chicks out of a 10 feet high hay loft. Mama says jump and they do. After they land, they run to her. Pardon me if I don't get all freaked out at the thought of them being a whole one foot off the ground. I know I'm being sarcastic and unkind, but I really think you are worrying over nothing.
I'll add this though. If your nest has a normal lip to hold the eggs and bedding in, there is extremely little chance of a chick falling out before Mama tells them to leave the nest. But if your nest does not have a reasonable lip on it, it is possible a chick could fall out. Not a high risk, but a risk. Unless you have had problems with eggs and bedding coming out, this is probably not a concern. But I am trying to deal with real life and some people have nests with no lip. Those baby chicks are extremely agile. They could probably get back in a nest 12" off the ground after they get a few hours age on them. But if it makes feel better, you can put a step that reduces that 12" to 6" or so. A paver, cinder block, something like that. Maybe just pile the bedding a little high right there if the other chickens won't scratch it back flat.
The target date for all chicken eggs is that they should hatch after 21 days of development. But that is just a target. In real life they can be a few days early or late. My last broody hatched 11 out of 11 and all of them were about two full days early. These were full-sized fowl from normal sized eggs. There are a whole lot of different things that can affect how early or late they hatch, even under a broody.
I'm guessing your Cochin is a bantam. Small eggs have a tendency to hatch a little earlier than larger eggs, but that is just a tendency. If you have thousands of eggs hatching, you might notice that. But with just a few hatching, I would not expect them hatching early to be a real high probability. They can be either early or late.
Chickens have been hatching with the flock for thousands of years. They are living animals and anything can happen, but they have been hatching with the flock for thousands of years. Mine hatch with the flock and I don't have problems. I won't lie to you and say you absolutely will not have problems. Sometimes people do, usually with the hens, not the rooster. But from my experience, the worry about something bad happening gets blown way out of proportion on this forum.
Heat is a real danger to chickens of all ages. Will that dog crate fit in your coop instead of in your greenhouse?
When a broody hatches, I normally wait until she decides to bring them off the nest, then put the broody and chicks in a specially prepared place for a couple of days just so the chicks learn to eat and drink without interference from the older hens. Many people don’t take that step to isolate them but just let them go. That’s just me being over-cautious. Then I open it up and let her raise them with the flock.
I would not move her yet. I would not take the chance of her rejecting the eggs and new nest. Wait until after the hatch and move them if you feel you must. I really think your odds of success are much greater if you do not move her.
Good luck!!